2007 iTest Problems/Problem 45
Revision as of 04:05, 16 June 2018 by Rockmanex3 (talk | contribs) (Solution to Problem 45 — grinding the base numbers)
Problem
Find the sum of all positive integers such that
, where
represent distinct base
digits,
.
Solution
Using the definition of base numbers, the equation can be rewritten as
To find the values of , use casework for values of
since
has the most influence on the value of
. Casework will be heavy, but a few tips can lighten the load. First, since
is one less than a multiple of
,
is congruent to
or
modulo
. Second, once
,
can not be higher for a given
. Third, use estimation to approximate the lower bound for a given
.
- If
, then
is just more than
. The first few values of
that work are
,
,
, and
. Testing each case,
when
and
.
- If
, then
is just more than
. The first few values of
that work are
,
,
, and
. Testing each case,
when
and
.
- If
, then
is just more than
. The first few values of
that work are
,
,
. Testing each case,
when
and
.
- If
, then
is just more than
. The first few values of
that work are
and
. After testing each case, no values of
work when
.
- If
, then
is just more than
. The first few values of
that work are
and
. After testing each case, once again, no values of
work when
.
- If
, then
is just more than
. The first few values of
that work are
and
. After testing each case, yet again, no values of
work when
.
In summary, the only possible values of are
and
, and the sum of the values equals
.
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 44 |
Followed by: Problem 46 | |
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